Gauss%20quadrature
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1: 3.5 Quadrature
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§3.5(v) Gauss Quadrature
… ►Gauss–Laguerre Formula
… ►§3.5(viii) Complex Gauss Quadrature
… ►a complex Gauss quadrature formula is available. …2: 35.10 Methods of Computation
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►Other methods include numerical quadrature applied to double and multiple integral representations.
See Yan (1992) for the and functions of matrix argument in the case , and Bingham et al. (1992) for Monte Carlo simulation on applied to a generalization of the integral (35.5.8).
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3: Bibliography G
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Werke. Band II.
pp. 436–447 (German).
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions.
J. Comput. Appl. Math. 139 (1), pp. 173–187.
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Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature.
ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.
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Calculation of Gauss quadrature rules.
Math. Comp. 23 (106), pp. 221–230.
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4: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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Highly Oscillatory Quadrature: The Story So Far.
In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.),
pp. 97–118.
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From Gauss to Painlevé: A Modern Theory of Special Functions.
Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.
5: 15.19 Methods of Computation
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►The Gauss series (15.2.1) converges for .
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►Large values of or , for example, delay convergence of the Gauss series, and may also lead to severe cancellation.
►For fast computation of with and complex, and with application to Pöschl–Teller–Ginocchio potential wave functions, see Michel and Stoitsov (2008).
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►Gauss quadrature approximations are discussed in Gautschi (2002b).
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►For example, in the half-plane we can use (15.12.2) or (15.12.3) to compute and , where is a large positive integer, and then apply (15.5.18) in the backward direction.
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6: 20.11 Generalizations and Analogs
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§20.11(i) Gauss Sum
►For relatively prime integers with and even, the Gauss sum is defined by … … ►
20.11.5
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►Similar identities can be constructed for , , and .
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7: 9.17 Methods of Computation
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►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979).
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►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8).
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►For quadrature methods for Scorer functions see Gil et al. (2001), Lee (1980), and Gordon (1970, Appendix A); but see also Gautschi (1983).
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8: 15.5 Derivatives and Contiguous Functions
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►The six functions , , are said to be contiguous to .
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15.5.11
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15.5.12
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►By repeated applications of (15.5.11)–(15.5.18) any function , in which are integers, can be expressed as a linear combination of and any one of its contiguous functions, with coefficients that are rational functions of , and .
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15.5.20
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9: 6.18 Methods of Computation
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►Quadrature of the integral representations is another effective method.
For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998).
For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968).
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►Power series, asymptotic expansions, and quadrature can also be used to compute the functions and .
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